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EE364a Homework 2 solutions
- quasiconvex
- quasiconcave
- probability
- semidefinite
- see.stanford.edu
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<strong>EE364a</strong>, Winter 2007-08 Prof. S. Boyd <strong>EE364a</strong> <strong>Homework</strong> 2 <strong>solutions</strong> 2.28 Positive semidefinite cone for n = 1, 2, 3. Give an explicit description of the positive semidefinite cone S n +, in terms of the matrix coefficients and ordinary inequalities, for n = 1, 2, 3. To describe a general element of S n , for n = 1, 2, 3, use the notation x 1 , [ ] x1 x 2 , x 2 x 3 ⎡ ⎤ x 1 x 2 x 3 ⎢ ⎥ ⎣ x 2 x 4 x 5 ⎦. x 3 x 5 x 6 Solution. For n = 1 the condition is x 1 ≥ 0. For n = 2 the condition is For n = 3 the condition is x 1 ≥ 0, x 3 ≥ 0, x 1 x 3 − x 2 2 ≥ 0. x 1 ≥ 0, x 4 ≥ 0, x 6 ≥ 0, x 1 x 4 −x 2 2 ≥ 0, x 4 x 6 −x 2 5 ≥ 0, x 1 x 6 −x 2 3 ≥ 0 and x 1 x 4 x 6 + 2x 2 x 3 x 5 − x 1 x 2 5 − x 6 x 2 2 − x 4 x 2 3 ≥ 0, i.e., all principal minors must be nonnegative. We give the proof for n = 3, assuming the result is true for n = 2. The matrix X = is positive semidefinite if and only if for all z. ⎡ ⎢ ⎣ ⎤ x 1 x 2 x 3 ⎥ x 2 x 4 x 5 ⎦ x 3 x 5 x 6 z T Xz = x 1 z 2 1 + 2x 2 z 1 z 2 + 2x 3 z 1 z 3 + x 4 z 2 2 + 2x 5 z 2 z 3 + x 6 z 2 3 ≥ 0 If x 1 = 0, we must have x 2 = x 3 = 0, so X ≽ 0 if and only if [ ] x4 x 5 ≽ 0. x 5 x 6 Applying the result for the 2 × 2-case, we conclude that if x 1 = 0, X ≽ 0 if and only if Now assume x 1 ≠ 0. We have x 2 = x 3 = 0, x 4 ≥ 0, x 6 ≥ 0, x 4 x 6 − x 2 5 ≥ 0. z T Xz = x 1 (z 1 +(x 2 /x 1 )z 2 +(x 3 /x 1 )z 3 ) 2 +(x 4 −x 2 2/x 1 )z 2 2+(x 6 −x 2 3/x 1 )z 2 3+2(x 5 −x 2 x 3 /x 1 )z 2 z 3 , 1
- Page 2 and 3: so it is clear that we must have x
- Page 4 and 5: 3 2 1 I II Along this line the func
- Page 6 and 7: (b) By inspection we have The deriv
- Page 8 and 9: f is quasiconcave because its super
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| , EE364a is the same as CME364a. This webpage contains basic course information; up to date and detailed information is on . For summer quarter 2023–2024, EE364a will be taught by . , or install .Course assistants: TA office hours and locations will be announced on Ed. The textbook is , available online, or in hard copy from your favorite book store. , due each Friday at midnight, starting the second week. We will use for homework submission, with the details on . We will have a late day policy on homeworks. Each student has one late day, i.e., you may submit one homework (except for homework 0) up to 24 hours late. Always reach out if you're facing unusual disruptions to your classwork. You are allowed, even encouraged, to work on the homework in small groups, but you must write up your own homework to hand in. Each question on the homework will be graded on a scale of {0, 1, 2}. . The format is a timed online 75 minute exam, at the end of the 3rd week. The midterm quiz covers chapters 1–3, and the concept of (DCP). . The format is a 24 hour take home exam, scheduled for the end of the last week of classes. You can take it during any 24 hour period over a multi-day period we'll fix later. We can arrange for you take it earlier (as a beta tester, and only if you really need to) but not later. The final exam will .Homework 20%, midterm 15%, final exam 65%. These weights are approximate; we reserve the right to change them later. Good knowledge of linear algebra (as in ) and probability. Exposure to numerical computing, optimization, and application fields helpful but not required; the applications will be kept basic and simple. You will use to write simple scripts, so basic familiarity with elementary Python programming is required. We will be supporting other packages for convex optimization, such as (Julia), (Matlab), and (R). In particular, the final exam will . Concentrates on recognizing and solving convex optimization problems that arise in applications. Convex sets, functions, and optimization problems. Basics of convex analysis. Least-squares, linear and quadratic programs, semidefinite programming, minimax, extremal volume, and other problems. Optimality conditions, duality theory, theorems of alternative, and applications. Interior-point methods. Applications to signal processing, statistics and machine learning, control and mechanical engineering, digital and analog circuit design, and finance. This course should benefit anyone who uses or will use scientific computing or optimization in engineering or related work (e.g., machine learning, finance). More specifically, people from the following departments and fields: Electrical Engineering (especially areas like signal and image processing, communications, control, EDA & CAD); Aero & Astro (control, navigation, design), Mechanical & Civil Engineering (especially robotics, control, structural analysis, optimization, design); Computer Science (especially machine learning, robotics, computer graphics, algorithms & complexity, computational geometry); Operations Research (MS&E at Stanford); Scientific Computing and Computational Mathematics. The course may be useful to students and researchers in several other fields as well: Mathematics, Statistics, Finance, Economics. . |
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EE364a, Winter 2007-08 Prof. S. Boyd EE364a Homework 2 solutions 2.28 Positive semidefinite cone for n = 1, 2, 3. Give an explicit description of the positive semidefinite cone Sn +, in terms of the matrix coefficients and ordinary inequalities, for n = 1, 2, 3. To describe a general element of Sn, for n = 1, 2, 3, use the notation x1, " x1 ...
The document provides solutions to homework problems about positive semidefinite cones, monotone nonnegative cones, level sets of convex and concave functions, and families of concave utility functions. For positive semidefinite cones, explicit descriptions are given for n=1,2,3 involving matrix coefficients and inequalities. For monotone nonnegative cones, it is shown to be a proper cone and ...
hw2 ee364a, winter prof. duchi ee364a homework solutions suppose is convex, and dom with show that for all solution. this is inequality with show that for all. Skip to document. University; High School. ... EE364a Homework 2 solutions. 3 Supposef:R→Ris convex, anda, b∈domfwitha < b.
Ee364a Homework 2 Solutions - Free download as PDF File (.pdf), Text File (.txt) or read online for free. ee364a homework 2 solutions
The matrix X = is positive semidefinite if and only if for all z. ⎡ ⎢ ⎣ ⎤ x 1 x 2 x 3 ⎥ x 2 x 4 x 5 ⎦ x 3 x 5 x 6 z T Xz = x 1 z 2 1 + 2x 2 z 1 z 2 + 2x 3 z 1 z 3 + x 4 z 2 2 + 2x 5 z 2 z 3 + x 6 z 2 3 ≥ 0 If x 1 = 0, we must have x 2 = x 3 = 0, so X ≽ 0 if and only if [ ] x4 x 5 ≽ 0. x 5 x 6 Applying the result for the 2 × 2 ...
EE364a Homework 4 solutions. 5 A simple example. Consider the optimization problem minimize x 2 + 1 subject to (x−2)(x−4)≤ 0 , with variablex∈R. (a) Analysis of primal problem. Give the feasible set, the optimal value, and the optimal solution. (b)Lagrangian and dual function. Plot the objectivex 2 + 1 versusx.
EE364a, Winter 2007-08 Prof. S. Boyd EE364a Homework 5 solutions 4.15 Relaxation of Boolean LP. In a Boolean linear program, the variable x is constrained to have components equal to zero or one: minimize cTx subject to Ax b xi ∈ {0,1}, i = 1,...,n. (1) In general, such problems are very difficult to solve, even though the feasible set is
Solution: The following Matlab script finds the approximate solutions using the heuristic methods proposed, as well as the exact solution. % illum_sol: finds approximate and exact solutions of % the illumination problem. clear all; % load input data illum_data; % heuristic method 1: equal lamp powers.
Ee364a Homework Solutions - Free download as PDF File (.pdf), Text File (.txt) or read online for free.
EE364a, Winter 2018-19 Prof. S. Boyd EE364a Homework 2 solutions 3.2 Level sets of convex, concave, quasiconvex, and quasiconcave functions. Some level sets of a function f are shown below. The curve labeled 1 shows {x | f (x) = 1}, etc. 3 2 1 Could f be . Solutions available. EE 364A.
EE364a is the same as CME364a. ... Weekly homework assignments, due each Friday at midnight, ... {0, 1, 2}. Midterm quiz. The format is a timed online 75 minute exam, at the end of the 3rd week. The midterm quiz covers chapters 1-3, and the concept of disciplined convex programming (DCP).
EE364a, Winter 2019-20 Prof. J. Duchi. EE364a Homework 7 solutions. 8 Euclidean cone containing given points. InRn, we define aEuclidean cone, with center directionc 6 = 0, and angular radiusθ, with 0≤θ≤π/2, as the set
EE364a, Winter 2007-08 Prof. S. Boyd EE364a Homework 2 solutions 2.28 Positive semidenite cone for n = 1, 2, 3. Give an explicit description of the positive semidenite cone Sn , in terms of the matrix coecients and ordinary inequalities, for + n = 1, 2, 3. Solutions available.
this holds for k = 2; you must show it for arbitrary k.) Hint. Use induction on k. Solution. This is readily shown by induction from the definition of convex set. We illustrate the idea for k = 3, leaving the general case to the reader. Suppose that x1,x2,x3 ∈ C, and θ1 + θ2 + θ3 = 1 with θ1,θ2,θ3 ≥ 0. We will show that y =
EE364a, Winter 2020-21 Prof. J. Duchi EE364a Homework 2 solutions 3.26 More functions of eigenvalues. Let λ1 (X) ≥ λ2 (X) ≥ · · · ≥ λn (X) denote the eigenvalues of a matrix X ∈ Sn . We have already seen several functions of the eigenvalues that are conve. Solutions available. CME 364A. Stanford University.
hw8 ee364a, winter prof. duchi ee364a homework solutions a8.6 newton method for approximate total variation total variation is based on the problem with the. Skip to document. University; High School. ... EE364a Homework 8 solutions. A8 method for approximate total variation de-noising. Total variation de-noising is based on the bi-criterion ...
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FC Saturn Moscow Oblast (Russian: ФК "Сатурн Московская область") was an association football club from Russia founded in 1991 and playing on professional level between 1993 and 2010. Since 2004 it was the farm club of FC Saturn Moscow Oblast. In early 2011, the parent club FC Saturn Moscow Oblast went bankrupt and dropped out of the Russian Premier League due to huge ...
EE364a Homework 6 solutions. 6.9 Minimax rational function fitting. Show that the following problem is quasiconvex: D = {(a, b) ∈ Rm+1 × Rn | q(t) > 0, α ≤ t ≤ β}. In this problem we fit a rational function p(t)/q(t) to given data, while constraining the denominator polynomial to be positive on the interval [α, β].
Elektrostal is a city in Moscow Oblast, Russia, located 58 kilometers east of Moscow. Elektrostal has about 158,000 residents. Mapcarta, the open map.
EE364a, Winter 2019-20 Prof. J. Duchi. EE364a Homework 1 solutions. 2 Level sets of convex, concave, quasiconvex, and quasiconcave functions. Which of the following setsS are polyhedra?
State Housing Inspectorate of the Moscow Region Elektrostal postal code 144009. See Google profile, Hours, Phone, Website and more for this business. 2.0 Cybo Score. Review on Cybo.
hw6 ee364a, winter prof. duchi ee364a homework solutions minimax rational function fitting. show that the following problem is quasiconvex: p(ti yi minimize max ... ri=rfixi , i= 1,... , k ri≥ 0 , i= 1,... , n ‖ci−cj‖ 2 ≤ri+rj, (i, j)∈ I. Alternative solution. Here is a nice alternative solution. Instead of explicitly working out ...